Measures of vote-seat disproportionality for incomplete data

Measures of the proportionality of distributions are used across disciplines. ‘Disproportionality indices’ represent an application in politics, comparing the seat allocation in parliaments to the votes expressed for political parties. Disproportionality in elections is particularly high when many votes are expressed for parties not entering parliament; in some elections such ‘wasted votes’ add up to two-digit vote percentages. However, ‘wasted votes’ for small parties below the electoral threshold, as well as votes for non-partisan candidates, are often not listed in detail in election statistics, and are instead lumped together in residual categories such as ‘Others’ or ‘Independents’. This can hide major discrepancies between vote and seat distributions. This risks introducing systematic bias into the analysis of elections. This paper discusses several theoretically based methods to estimate indices of disproportionality for incomplete data, based on different theoretical scenarios concerning the distribution of votes and seats, and inspired by Taagepera’s method of ‘logical boundaries’. Empirical tests, relying on a dataset of 735 parliamentary elections worldwide, show that residual categories substantially affect indices of disproportionality. Several methods can considerably improve the measurement validity compared to the frequently used ‘naive’ procedures.